Breaking Down All Things Algebra Unit 3 Homework 2: The Untold Side – A Step-by-Step Guide

This guide will walk you through “The Untold Side” homework from All Things Algebra’s Unit 3 (likely focusing on systems of equations or inequalities). We'll break down the problems, explain the concepts, and provide troubleshooting tips so you can confidently complete the assignment.

Prerequisites:

Before diving into this homework, make sure you have a solid understanding of the following:

  • Basic Algebra: Familiarity with variables, constants, coefficients, and operations like addition, subtraction, multiplication, and division involving algebraic expressions.

  • Graphing Linear Equations: Knowing how to plot points on a coordinate plane and graph linear equations (lines) using slope-intercept form (y = mx + b) or standard form (Ax + By = C).

  • Solving Linear Equations: Being able to isolate variables to solve for their values in single-variable equations.

  • Systems of Equations (Introduction): A basic understanding of what a system of equations is (two or more equations with the same variables) and the goal of finding a solution that satisfies all equations simultaneously.

  • Inequalities (Introduction): A basic understanding of what an inequality is (a statement that compares two expressions using symbols like <, >, ≤, or ≥).

    • Tools You'll Need:

  • Pencil and Eraser: For writing and correcting your work.

  • Graph Paper (Optional but Recommended): Helps with accurate graphing.

  • Ruler or Straightedge: Essential for drawing straight lines on your graphs.

  • Calculator (Optional but Helpful): Can speed up calculations, especially with larger numbers.

  • Textbook or Class Notes: Refer to these for definitions, examples, and formulas.

  • Online Graphing Calculator (Desmos is a good choice): Useful for checking your work and visualizing graphs.

  • This Guide!: Your step-by-step companion.

    • Numbered Steps for Tackling "The Untold Side":

    While the specific questions in "The Untold Side" will vary, the general approach to solving systems of equations and inequalities remains consistent. Here's a breakdown of common problem types and how to solve them:

    1. Identifying the Problem Type:

  • Systems of Equations: Look for two or more equations with the same variables (usually x and y). The goal is to find the (x, y) coordinate(s) that satisfy *all* equations.

  • Systems of Inequalities: Look for two or more inequalities with the same variables. The goal is to find the region on the graph where *all* inequalities are true.

  • Word Problems: These will be presented as real-world scenarios that you need to translate into mathematical equations or inequalities.

    • 2. Solving Systems of Equations:

    There are three primary methods for solving systems of equations:

  • a) Graphing:

    • 1. Graph each equation: Convert each equation into slope-intercept form (y = mx + b) if necessary. Plot the y-intercept (b) on the y-axis, and then use the slope (m) to find other points on the line. Draw the line through these points.
    2. Identify the Intersection: The solution to the system is the point where the lines intersect. Write the coordinates of this point (x, y).
    3. Check Your Solution: Substitute the x and y values into *both* original equations. If both equations are true, you've found the correct solution.

  • b) Substitution:

    • 1. Solve for a Variable: Choose one of the equations and solve for one of the variables (either x or y).
    2. Substitute: Substitute the expression you found in step 1 into the *other* equation. This will leave you with an equation with only one variable.
    3. Solve for the Remaining Variable: Solve the equation from step 2 for the remaining variable.
    4. Substitute Back: Substitute the value you found in step 3 back into either of the original equations (or the equation from step 1) to solve for the other variable.
    5. Write Your Solution: Write your solution as an ordered pair (x, y).
    6. Check Your Solution: Substitute the x and y values into *both* original equations.

  • c) Elimination (Addition/Subtraction):

    • 1. Line Up Variables: Make sure the x and y terms are aligned in both equations.
    2. Multiply (if necessary): Multiply one or both equations by a constant so that the coefficients of either x or y are opposites (e.g., 3x and -3x).
    3. Add the Equations: Add the two equations together. This will eliminate one of the variables.
    4. Solve for the Remaining Variable: Solve the equation from step 3 for the remaining variable.
    5. Substitute Back: Substitute the value you found in step 4 back into either of the original equations to solve for the other variable.
    6. Write Your Solution: Write your solution as an ordered pair (x, y).
    7. Check Your Solution: Substitute the x and y values into *both* original equations.

    3. Solving Systems of Inequalities:

  • a) Graph each inequality:

    • 1. Convert to Slope-Intercept Form: Rewrite each inequality in slope-intercept form (y < mx + b, y > mx + b, y ≤ mx + b, or y ≥ mx + b).
    2. Graph the Boundary Line: Graph the line y = mx + b. Use a solid line if the inequality includes "or equal to" (≤ or ≥), and a dashed line if it doesn't (< or >).
    3. Shade the Correct Region: Choose a test point (e.g., (0, 0)) that is *not* on the line. Substitute the x and y values of the test point into the original inequality.
    * If the inequality is true, shade the region that contains the test point.
    * If the inequality is false, shade the region that does *not* contain the test point.

  • b) Identify the Solution Region: The solution to the system of inequalities is the region where the shaded areas of *all* inequalities overlap. This region represents all the (x, y) coordinates that satisfy all inequalities simultaneously.

    • 4. Tackling Word Problems:

  • a) Read Carefully: Understand the problem thoroughly. Identify what you are trying to find.

  • b) Define Variables: Assign variables (e.g., x and y) to represent the unknown quantities.

  • c) Translate into Equations/Inequalities: Write equations or inequalities that represent the relationships described in the problem. This is often the trickiest part. Look for keywords like "sum," "difference," "is," "greater than," "less than," etc.

  • d) Solve the System: Use one of the methods described above to solve the system of equations or inequalities.

  • e) Answer the Question: Make sure you answer the question that was asked in the problem. Include appropriate units.

    • Troubleshooting Tips:

  • Double-Check Your Work: A simple arithmetic error can throw off the entire solution.

  • Use a Graphing Calculator: Use Desmos or another graphing calculator to check your graphs and solutions.

  • Rewrite Equations: Sometimes, rewriting an equation in a different form can make it easier to solve.

  • Ask for Help: Don't be afraid to ask your teacher, classmates, or a tutor for help if you're stuck.

  • Practice, Practice, Practice: The more you practice solving systems of equations and inequalities, the easier it will become.

Short Summary:

"The Untold Side" homework from All Things Algebra Unit 3 is likely focused on mastering the skills needed to solve systems of equations and inequalities. This guide provides a structured approach to tackling these problems, covering everything from identifying problem types and selecting appropriate solution methods to troubleshooting common errors. By following these steps and practicing consistently, you'll be well-equipped to conquer this homework and build a solid foundation in algebra. Good luck!